Mathematics > Number Theory

Abstract: We study Piatetski-Shapiro sequences $(\lfloor n^c\rfloor)_n$ modulo m, for
non-integer $c >1$ and positive $m$, and we are particularly interested in
subword occurrences in those sequences. We prove that each block $\in\{0,1\}^k$
of length $k < c + 1$ occurs as a subword with the frequency $2^{-k}$, while
there are always blocks that do not occur. In particular, those sequences are
not normal. For $1<c<2$, we estimate the number of subwords from above and
below, yielding the fact that our sequences are deterministic and not morphic.
Finally, using the Daboussi-Kátai criterion, we prove that the sequence
$\lfloor n^c\rfloor$ modulo m is asymptotically orthogonal to multiplicative
functions bounded by $1$ and with mean value $0$.